Chapter-Development of EDM, PPT, Semester, Engineering - Computer Science Engineering (CSE) PDF Download

Development of Empirical Dynamic Models from Step Response Data

Black box models

• step response easiest to use but may upset the plant manager (size of input change? move to new steady-state?)
• other methods
  • impulse - dye injection, tracer
  • random - PRBS (pseudo random binary sequences)
  • sinusoidal - theoretical approach
  • frequency response - modest usage (incl. pulse testing)
  • on-line (under FB control)

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Some processes too complicated to model using physical principles

•  material, energy balances
•  thermodynamics
•  physical properties (often unknown)
•  flow dynamics

Example 1: distillation column

50 plates

• For a 50 plate column, dynamic models have many ODEs that require model simplification;  and physical properties must  be known; e.g., HYSYS
• black box models (only good for fixed operating conditions) but requires operating plant (actual data)
• theoretical models must be used prior to plant construction or for new process chemistry

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• Need to minimize disturbances during a plant test

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Simple Process Models

1st order system with gain K, dead time  θ and time constant τ; 3 parameters to be fitted.

Step response:

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For a 1st order model, we note the following characteristics (step response)

(1)  The response attains 63.2% of its final response at one time constant (t = τ+θ ).

(2)  The line drawn tangent to the response at maximum slope (t = θ) intersects the 100% 

line at (t = τ+θ ). [see Fig. 7.2]

K is found from the steady state response for an input change magnitude M.

There are 4 generally accepted graphical techniques for determining first order system parameters τ, θ:

1. 63.2% response

2. point of inflection

3. S&K method

4. semilog plot  

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Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide

Inflection point hard to find with noisy data.

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S & K Method for Fitting FOPTD Model

• Normalize step response

(t = 0, y = 0; t →∞, y = 1)

• Use 35 and 85% response times (t1 and t2)

θ = 1.3 t1 – 0.29 t2

τ = 0.67 (t2 – t1)

(based on analyzing many step responses)

K found from steady state response

• Alternatively, use Excel Solver to fit θ and τ using y (t) = K [1– e –(t-θ)/τ] and data of y vs. t

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Fitting an Integrator Model to Step Response Data

In Chapter 5 we considered the response of a first-order process to a step change in input of magnitude M:

For short times, t < τ, the exponential term can be approximated by

so that the approximate response is:

(straight line with slope of y1(t=0))

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is virtually indistinguishable from the step response of the integrating element

matches the early ramp-like response to a step change in input.

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Figure 7.10. Comparison of step responses for a FOPTD model (solid line) and the approximate integrator plus time delay model (dashed line).

Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide

Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide

Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide

Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide

Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide

Smith’s Method

20% response:    t20 = 1.85

60% response:    t60 = 5.0

t20 / t60 = 0.37

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Using Excel Solver to Fit Transfer Function Models

• use y (data) vs. y (predicted)

• column 1 is data (taken at different times), or y1

• column 2 is model prediction (same time values as above), or y2

• target cell is Σ (y1 - y2)2 , to be minimized

• specify parameters to be changed in reference cells (e.g. τ1 = 1, τ2 = 2)

• open solver dialog box to check settings

• click on < solve > (calls optimization program)

Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide

Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide

Development of Empirical Dynamic  Models from Step Response Data---------------Next Slide